Arnason, G., 1958: A convergent method for solving the balance equation. J. Meteorol., 15, 220−225, https://doi.org/10.1175/1520-0469(1958)015<0220:ACMFST>2.0.CO;2.
Asselin, R., 1967: The operational solution of the balance equation. Tellus, 19, 24−32, https://doi.org/10.3402/tellusa.v19i1.9725.
Bessho, K., M. DeMaria, and J. A. Knaff, 2006: Tropical cyclone wind retrievals from the advanced microwave sounding unit: Application to surface wind analysis. J. Appl. Meteorol. Climatol., 45, 399−415, https://doi.org/10.1175/JAM2352.1.
Bijlsma, S. J., and R. J. Hoogendoorn, 1983: A convergence analysis of a numerical method for solving the balance equation. Mon. Wea. Rev., 111, 997−1001, https://doi.org/10.1175/1520-0493(1983)111<0997:ACAOAN>2.0.CO;2.
Bolin, B., 1955: Numerical forecasting with the barotropic model. Tellus, 7, 27−49.
Bolin, B., 1956: An improved barotropic model and some aspects of using the balance equation for three-dimensional flow. Tellus, 8, 61−75, https://doi.org/10.1111/j.2153-3490.1956.tb01195.x.
Boyd, J. P., 1999: The Devil's invention: Asymptotic, superasymptotic and hyperasymptotic series. Acta Applicandae Mathematicae, 56, 1−98, https://doi.org/10.1023/A:1006145903624.
Bring, A., and E. Charasch, 1958: An experiment in numerical prediction with two non-geostrophic barotropic models. Tellus, 10, 88−94, https://doi.org/10.3402/tellusa.v10i1.9216.
Bushby, F. H., and V. M. Huckle, 1956: The use of a stream function in a two-parameter model of the atmosphere. Quart. J. Roy. Meteor. Soc., 82, 409−418, https://doi.org/10.1002/qj.49708235404.
Cao, J., and Q. Xu, 2011: Computing streamfunction and velocity potential in a limited domain of arbitrary shape. Part II: Numerical methods and test experiments. Adv. Atmos. Sci., 28, 1445−1458, https://doi.org/10.1007/s00376-011-0186-5.
Charney, J., 1955: The use of the primitive equations of motion in numerical prediction. Tellus, 7, 22−26, https://doi.org/10.1111/j.2153-3490.1955.tb01138.x.
Charney, J. G., and M. E. Stern, 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci., 19, 159−172, https://doi.org/10.1175/1520-0469(1962)019<0159:OTSOIB>2.0.CO;2.
Courant, R., and D. Hilbert, 1962: Methods of Mathematical Physics. Vol. II, Interscience, 830 pp.
Kieu, C. Q., and D.-L. Zhang, 2010: A piecewise potential vorticity inversion algorithm and its application to hurricane inner-core anomalies. J. Atmos. Sci., 67, 2616−2631, https://doi.org/10.1175/2010JAS3421.1.
Kuo, H. L., 1959: Finite-amplitude three-dimensional harmonic waves on the spherical earth. J. Meteorol., 16, 524−534, https://doi.org/10.1175/1520-0469(1959)016<0524:FATDHW>2.0.CO;2.
Liao, T. H., and T. T. Chow, 1962: On the method for solving the balance equation in finite difference form. Acta Meteorologica Sinica, 32, 224−231, https://doi.org/10.11676/qxxb1962.020. (in Chinese with English abstract
Miyakoda, K., 1956: On a method of solving the balance equation. J. Meteor. Soc. Japan, 34, 364−367, https://doi.org/10.2151/jmsj1923.34.6_364.
Schubert, W. H., R. K. Taft, and L. G. Silvers, 2009: Shallow water quasi-geostrophic theory on the sphere. Journal of Advances in Modeling Earth Systems, 1, 2, https://doi.org/10.3894/JAMES.2009.1.2.
Shuman, F. G., 1955: A method for solving the balance equation. Technical Memorandum No. 6, Joint Numerical Weather Prediction Unit, 12 pp.
Shuman, F. G., 1957: Numerical methods in weather prediction: I. The balance equation. Mon. Wea. Rev., 85, 329−332, https://doi.org/10.1175/1520-0493(1957)085<0329:NMIWPI>2.0.CO;2.
Southwell, R. V., 1946: Relaxation Methods in Theoretical Physics. Clarendon Press, 248 pp.
Velden, C. S., and W. L. Smith, 1983: Monitoring tropical cyclone evolution with NOAA satellite microwave observations. J. Appl. Meteorol. Climatol., 22, 714−724, https://doi.org/10.1175/1520-0450(1983)022<0714:MTCEWN>2.0.CO;2.
Wang, X. B., and D.-L. Zhang, 2003: Potential vorticity diagnosis of a simulated hurricane. Part I: Formulation and quasi-balanced flow. J. Atmos. Sci., 60, 1593−1607, https://doi.org/10.1175/2999.1.
Xu, Q., 1994: Semibalance model—connection between geostrophic-type and balanced-type intermediate models. J. Atmos. Sci., 51, 953−970, https://doi.org/10.1175/1520-0469(1994)051<0953:SMBGTA>2.0.CO;2.
Xu, Q., and J. Cao, 2012: Semibalance model in terrain-following coordinates. J. Atmos. Sci., 69, 2201−2206, https://doi.org/10.1175/JAS-D-12-012.1.
Xu, Q., J. Cao, and S. T. Gao, 2011: Computing streamfunction and velocity potential in a limited domain of arbitrary shape. Part I: Theory and integral formulae. Adv. Atmos. Sci., 28, 1433−1444, https://doi.org/10.1007/s00376-011-0185-6.
Zhu, T., D.-L. Zhang, and F. Z. Weng, 2002: Impact of the advanced microwave sounding unit measurements on hurricane prediction. Mon. Wea. Rev., 130, 2416−2432, https://doi.org/10.1175/1520-0493(2002)130<2416:IOTAMS>2.0.CO;2.